Friday, November 2, 2012

Political Independence

I’m
getting rather perturbed with this constant stream of naysayers, who claim that
voting is unimportant. One argument trotted out on a consistent basis goes
something like this. A liberal individual voting in a very conservative county,
in a severely conservative state, has very little likelihood of swaying
national, state-wide, and/or county-wide elections due to the sheer numbers
game. One vote out of millions will do nothing to change an election. Many
people counter with arguments about one’s civic duty to vote, or how people in
other parts of the world are willing to die for the right to vote. All of these
arguments are persuasive on moral and philosophical grounds, but if we are
dealing with avid utilitarian’s, who are hell bent on claiming that the
likelihood one’s vote matters is low, none of these arguments prove that the
ends justify the means. What all these counter-arguments miss, however, is that
there is a pernicious, downright improper assumption running through these
anti-voting epithets. That problem is of course the assumption of independence.
When I say independence, I’m talking about probabilistic independence, having
no thoughts of anything more patriotic or interesting.

Let’s
take a step back and talk about this sort of independence and then it hopefully
will become clear why these anti-voting authors’ assumption does not bode well
for how the real world works. All independence says is given two events, A and
B, knowing that A has already happened, the likelihood of B happening has not
changed, and vice versa. Thus, two events being independent translate into two
events that have no relationship to one another. Take this simple example. If I
go home and decide to cook myself dinner, the likelihood that my next door
neighbor, who I have never met, cooks dinner is unchanged. This is for both
physical reasons, we don’t share a living space, and social reasons, we don’t have
any means of communicating with one another. However, when I decide to cook
dinner, my roommate’s likelihood of cooking dinner does change. Either for a
physical reason, I’m taking up the stove so he can’t cook, or a social reason,
he sees me cooking dinner and decides he can just take some of my food. So we
would say that the likelihoods of me and my next door neighbor cooking dinner
are independent, while the likelihoods of me and my roommate cooking dinner are
dependent.

Hopefully,
one can see where I’m going with this. All those authors who decry about the
likelihood one’s vote matters assume that a person’s choice to vote is
independent of everyone else’s choice to vote. If everyone’s decision to vote
is independent, then the likelihood that I vote having any sort of meaning on
the election is rather small. However, if my decision to vote persuades another
family member, or friend in my social network, to vote, and their decision also
causes another to vote, then the cascading dependencies can really make the
numbers a bit more favorable. Let’s go to a simple statistical example
I’ve concocted. In the table below, we can see the joint distributions between
two people, person A and person B voting, where they have dependent
probabilities:

VOTER B

P( Vote)

P(Not Vote)

VOTER A

P( Vote)

0.5

0.2

P(Not Vote)

0.05

0.25

The table is very simple, so bear with me as I walk you
through it. For example, sum across both rows and you get the marginal
probabilities of A voting and not voting:

You’ll
notice if we sum these two numbers together we get 1, and that makes sense
because Voter A can only vote or not vote, thus they will do either/or 100% of
the time. We can now ask questions regarding conditional probabilities. Given
that voter B has voted, what is the probability that voter A will also vote? To
answer this, we restrict ourselves to only the first column and see that voter
A votes with P(Voter A Votes) = .5 and voter A does not vote with P(Voter A Not
Votes) = .05. However, the sum of these two are only .55 . So to find the
probability that voter A votes when voter B has voted, we simply divide P(Voter
A Votes) = .5 by our restricted sample space, the marginal probability of B
voting, which is equal to .55 . So we have P(Voter A Votes| Voter B Votes) = .5/.55 = .91. We can now verify these two
voters are dependent because the likelihood that A votes changes whenever B
decides to vote or not vote. So doing the same process we would find P(Voter A
Votes | Voter B Not Votes) = .2/.45 =
.44 . Now things are getting interesting, given that B votes the probability
that A votes nearly doubles. This means that voter B’s decision to vote also
increased the likelihood that voter A would vote by a factor of 2.

Now
let’s complicate things a little bit more. Let’s say that we have a pool of 100
voters, what is the likelihood that a single voter could sway the election? Well if we assume that every person’s vote is independent that is very easy,
it’s merely 1 in 100, or .01. However, using our numbers from before let’s say
voter B decides to vote before voter A. So voter B has a probability of .01 of
swinging the election, but because her decision to vote has that added benefit
of increasing the probability that A votes by .91 - .44 = .47, the likelihood
that voter B’s vote is the decider is now .01 + .01*.47 = .0147.

What this
example is trying to illustrate is that if there are externalities to voting
then the likelihood one’s vote matters can start to creep upwards. Given that our
social networks are increasing faster and faster with the plethora of social media,
these small spillovers start to add up. However, is a person’s choice to vote
independent of everyone else’s choice to vote? Recent work by Betsy Sinclair at
the University of Chicago, which looked at political canvassers in Los Angeles,
seems to suggest that the independence assumption is incorrect. Networks
influence political activity via social pressure to conform. Previous research
had found that people in social networks where with high rates of political
activity conform to social pressure and are more likely to acquiesce and vote.
The important point in Sinclair’s work is that the messenger matters, if the
politically active person trying to gin up votes is not in the same group as
the person they are persuading then the effects on political behavior is small.
However, if both people are members of similar groups, then the effects may be
very large. For hermits that lack any social networks, the likelihood that
there vote matters is infinitesimally small, however, for those people
inculcated in vast networks of friends, families, and co-workers, their vote
may make more of a difference than pundits make you believe.